Theorem concerning analytic continuation

A. I. Markushevich obtained the following representation of a function in its holomorphicity star with a sequence {m _v }, for which m _v+1/m _v: Here it is proved that this condition is necessary; more precisely, . This result is derived from certain properties of over-convergent power series.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 57–62, July, 1971.The author wishes to thank his scientific director A. I. Markushevich for his advice concerning this work.     

Unique continuation for operatorswith partially analytic coefficients

A general unique continuation result for partial differential operators with partially analytic coefficients was obtained in Tataru (1995); however, certain technical assumptions were used there. A part of these assumptions were elliminated independently by Hörmander (1985), and Robbiano and Zuily (preprint). The aim of this note is to remove the remaining technical restriction and, at the same time, to provide a simple proof for the entire result.     

A remark on analytic continuation

A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.

     

Quad mesh generation for k-sided faces and hex mesh generation for trivalent polyhedra

Two mesh generation methods are presented in this paper: one for quadrilateral meshing of faces with convex vertices but an arbitrary number of sides, the other hexahedral meshing of polyhedra with convex edges, vertices connected to three edges and an arbitrary number of faces each of which has at least three sides. Efficient mapping methods for determining nodal positions are developed based on an extension to transfinite mapping and it is shown that these are identical to those obtained by iterative isoparametric smoothing.     

An optimal filtering method for stable analytic continuation

In this paper, we consider the problem of numerical analytic continuation of an analytic function f(z)=f(x+iy) on a strip domain Ω⁺={z=x+iy∈C∣x∈R,0<y<y₀}, where the data is given approximately only on the real axis y=0. This problem is severely ill-posed: the solution does not depend continuously on the given data. A novel method (filtering) is used to solve this problem and an optimal error estimate with Hölder type is proved. Numerical examples show that this method works effectively.     

Distributions and analytic continuation of Dirichlet series

This is the second in a series of three papers; the other two are “Summation Formulas, from Poisson and Voronoi to the Present” [Progr. Math. 220 (2004) 419-440] and “Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)” (preprint). The first paper is primarily expository, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on . The present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)×GL(1) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function, degree 1 and degree 2 L-functions, the converse theorem for GL(2), and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities.     

On the continuation of analytic sets

Mathematische Annalen -     

The analytic continuation of hyperbolic space

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantages of this new geometric model as well as some of its applications.     

Anisotropic mesh generation methods based on ACVT and natural metric for anisotropic elliptic equation

Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features.Defning an appropriate metric tensor and designing an efcient algorithm for anisotropic mesh generation are two important aspects of the anisotropic mesh methodology.In this paper,we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems.We provide an algorithm to generate anisotropic meshes under the given metric tensor.We show that the inverse of the anisotropic difusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects:better discrete algebraic systems,more accurate fnite element solution and superconvergence on the mesh nodes.Various numerical examples demonstrating the efectiveness are presented.